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A square medieval castle on a square island is under siege. All around the castle there is a square moat 10 meters wide. Due to a regrettable miscalculation the raiders have brought footbridges, which are only 9.5 meters long. The invaders cannot abandon their campaign and return empty-handed.

How can the assailants resolve their predicament?

This old topic is locked since it was answered many times. You can check solution in the Spoiler below.

You can put one foot-bridge over one corner (thus a triangle is created). Then from the middle of this foot-bridge lay another foot-bridge to the edge (corner) of the castle. You can make a few easy equations confirming that this is enough.

A square medieval castle on a square island was under siege. All around the island, there was a 10 metre wide water moat. But the conquerors could make foot-bridges only 9.5 metres long. Nevertheless a wise man was able to figure out how to get over the water. What do you think was his advice?

(There's a place on the other side to put the bridge against, not just a sheer wall. the water moat has square corners - that section of the moat is about 14.1 metres wide.)

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I've got an idea: swim. Or put floating barrels on the water with the 9.5m planks firmly attached so they float atop the barrels. By attaching 10 or 20 in a row, you could safely walk across the middle without falling over.

Another solution I've actually done:

Place one board so it's sticking out 1.5 meters over the edge. It has 8 meters still on flat ground. Now, have 1 man stand on the end furthest from the moat. Because of mechanical advantage, you can put the weight of ~5.33 men on the other end of the board (8 / 1.5 ~= 5.33 ). Carry a board vertically 1 meter out, then drop the top end so it swings down and lands on the opposite edge. Set the close edge so .5 meters of the second board is resting on the first board, and there will be .5 meters resting on the opposite side, with the remaining 8.5 meters bridging the gap. I'm not sure of the exact physics, but I know that the maximum mechanical disadvantage for the guy holding the board down is 10:8, or 1.25, and that would occur when someone was standing at the very opposite edge of the moat. So 2 guys on the end away from the castle would be sufficient for 1 guy to cross the bridge.

Once the first guy is across, you have a few options:

1. Send a second guy across (total of 4 guys), reverse the boards (so the far board has 1.5m sticking over the moat and the near board covers the remaining 8.5 meters), then walk the remaining 2 guys across, bringing the second board with them. (This is the method I used, because the objective was to cross the gap and bring the boards with us so we could cross the remaining gaps.)

2. Pull both boards so that .75 meters are sticking over the moat and use a 3rd board to cover the remaining 8.5 meters. Now, you have a worst case advantage of 1.75:1, because you have 8.75 meters on land, and if the load is more than 5 meters from your edge, the other board is taking the extra load. (This is basically the same setup as Skumbag's, except I'm using weight instead of lashing to keep things from falling.)

3. Pull both boards so 5 meters is over the moat and the remaining 4.5 meters is on dry land on either edge. The disadvantage is now 1.11:1, so 1 big guy holding some heavy-ish stuff could get smaller guys across.

You could also put something very heavy on the end to support more than 2 people crossing at a time (such as an anvil or large tank filled with water). Your wieght-over-the-water would ultimately be limited by how much stress the boards could take, but you could always roll heavier objects across multiple boards at once to distribute the load.

The disadvantage of my method is the requirement to leave something heavy on the ends while people are crossing, but it has the advantage that the army can breach the gap anywhere along the moat. I would guess that for any reasonable army, the advantage would far outweigh the disadvantage. Of course, I'd also guess that a reasonable army could make 10+ meters boards.

Note: Umm. I made a mistake in my counting somewhere, and don't feel like fixing it at the moment. However, the concept remains valid. Edit: the mistake is on the picture, frame 4. It should read "8.5 m", not "7.5 m".

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lol, I LOVE IT. That was not only a very wise discovery on how to gain access to the other side, but a cool little diagram you built to explain it better.. I like the art work..lol Plus, your way does not leave any Gaps and is a straight shot across. I suppose Rookies answer does not leave any gaps either, but would require you to walk in from the side and turn. Either way, Both very cool answers... I, on the other hand, would hire a Cuban to carry me across!

(I can say that, I'm Cuban..lol)

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a simple solution is to build the 9.5 foot bridge and jump the other .5 meters (or 1.5 feet) Unless of course we are assuming the bridge will not hold without a base on the other side. This could easily be solved with a simple rope support system; attached the ropes to the end and periodically along the edge of the bridge to support it. The ropes would be run up to a tree, that happens to be near by, and then fastened to the ground.

-Remember today is the tomorrow you worried about yesterday.

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well the problem with jumping is... the bridge would fall over as soon as you reached the end... the more logical sence would be like in a^2+b^2=c^2... would be to turn it on its side so the diagonals were touching both sides... assuming of course these were soldiers and not tight rope walkers and the bridge had width

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You guys are missing a key aspect of this scenario...the castle is on an island. If you just make a canal linking to the ocean, the moat will drain itself and you won't even need to bother with foot bridges.

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or you could have someone hold down the first, say, 3 meters on the ground to suport the 6.5 meters hanging out over the moat. Then there would only be 3.5 meters (about 10 feet) to longjump, easily. And if someone couldnt, they could just bounce on the board because it would be springy, and then their jump would be much more powerful anyway. Easy.

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Well, after reading the responses, I think the question can close some of the loopholes by a little rephrasing.

Some kids were wandering in the hills when they came across a square medieval castle on a square island. 1 meter from the edge of the walls was a moat - 10 metres wide - filled with foul-smelling water.

(Naturally, the moat has square corners, and the distance from an outer corner to its inner corner is about 14.1 metres wide.)

They found a couple of 9.5 metre planks, but otherwise had no tools to cross the moat. Nevertheless, they figured out a way across. What do you think they did?

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for those of you who suggested putting 1 unit of bridge diagonally across the bridge and then placing the other 8.5 unit length to complete the bridge on the corner have forgotten a simple process of calculation. Pythagorean therum suggests that the corner that we now are attempting to cross is no longer 10 units of length, but 14.1 units of length. and the diagonal idea only acconts for 0.866 unites of length, therefore, there is still 13.23 units of lenth to cover... and guess what, technology has not been updated... they can still only build 9.5M bridges, and i think the question asks us to assume that they only brought enough wood and nails for one bridge

silly them

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First, about the post above mine, I don't know if I'm overlooking something, but why do you say 8.5 unit thength of bridge is being used to complete the diagonal bridge when the bridges are 9.5 and not 8.5 meters long?

I tried using equations with the two boards at the corner thing and found that there is still .1 meters that aren't acounted for from the outer corner to the inner corner. I'll show my work 'cause with a .1 unnaccounted answer I think I got something backwards and that should be the extra. So see if you can find my fault, please.

Firstly, once they are laid, there will need to be at least 4.6 meters that the bridge forming the triangle will need to cover from the corner to it's central point.

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Also, Assuming it's an army andnot kids tht stuimbled upon it, then you didn't realize that he said that if jumping wouldn't work because of the board falling then you could fasten it to a tree with rope. Which would work, assuming you're in track and can jump that far. But, even if you can't do that then you could still put the other bridge accross the rest now that the first is fastened to the tree and can stand the weight. If it were kids that found it then, no, it would work. But We are supposed to be talking about an army, I'm sure they'll have a bit of rope with them. Finally, the question never states one bridge. If it did, then there would be no way except to climb up to stand on the very end and jumping as it falls into the moat and you lose it. Just look at the pics from earlier (one of the first posts, it has five or six frames) and imagine having someone stand on the end of the bridge in the third frame but without the board on the ground while they raise it and push it into the moat so the person could jump off onto the other side. This gets one person over. Completely useless. So saying that it implies a single bridge can't be right because then the above would be all you could do.

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but why do you say 8.5 unit thength of bridge is being used to complete the diagonal bridge when the bridges are 9.5 and not 8.5 meters long?

from the way i read it i assumed that the bridge can only be 9.5 M and only one bridge can be made. Therefore if one 1M was used on the diagonal, only 8.5M of bridge would be able to be made with the remaning material

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You can put one foot-bridge over one corner (thus a triangle is created). Then from the middle of this foot-bridge lay another foot-bridge to the edge (corner) of the castle. You can make a few easy equations confirming that this is enough.

going of what the soultion claims... does he mean 1 ft, or 1M?

or, does he say 1 ft to imply 1/3 M?

thats why i started saying 'unit length' as opposed to M or ft

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You can put one foot-bridge over one corner (thus a triangle is created). Then from the middle of this foot-bridge lay another foot-bridge to the edge (corner) of the castle. You can make a few easy equations confirming that this is enough.

going of what the soultion claims... does he mean 1 ft, or 1M?

or, does he say 1 ft to imply 1/3 M?

thats why i started saying 'unit length' as opposed to M or ft

unit length = metres

footbridge = a bridge designed for pedestrians

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but why do you say 8.5 unit thength of bridge is being used to complete the diagonal bridge when the bridges are 9.5 and not 8.5 meters long?

from the way i read it i assumed that the bridge can only be 9.5 M and only one bridge can be made. Therefore if one 1M was used on the diagonal, only 8.5M of bridge would be able to be made with the remaning material

I see what you meant by it now.

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What everyone seems to be missing is that the equations tell us that the solution "should" work. Practically, it would not work. The bridge to the castle would have to be supported by the land and the other bridge. According to the equations, the "distance" from the castle to the other bridge is exactly 9.5. At least that's what I see.